Lemma 2.7. Let Y be a beta 2,1 random variable and X = ǫY +1−ǫ1−Y where ǫ is independent of Y and such that Pǫ =
1 = Pǫ = −1 = 12. Then X is uniform on [0, 1]. Furthermore, if W is integrable and independent of ǫ, then we have E[W |X ] = X gX +
1 − X g1 − X where g y = E[W |Y = y].
We also get, thanks to the above Lemma that: E
[τ|X = x] = −2 x logx + 1 − x log1 − x ,
and 15
E [τ
2
|X = x] = 8 x
logx + 1 − x log1 − x − x Z
1 x
log1 − z z
dz 16
−1 − x Z
1 1−x
log1 − z z
dz .
2.5 Number of individuals present which will become MRCA
We keep notations from Sections 2.1 and 2.3. We set Z = Z
d
G∗
the number of individuals living at time d
G
∗
which will become MRCA of the population in the future. Let L = Ld
G
∗
+ 1 and L
, L
1
, . . . , L
Z
= L d
G
∗
, . . . , L
Z
d
G
∗
be the levels of the fixation curves at the death time of G
∗
. Recall notations from Section 2.2. The following Lemma and Proposition 2.4 characterize the joint
distribution of Y, Z, L, L
1
, . . . , L
Z
.
Lemma 2.8. Conditionally on L, Y the distribution of Z, L
1
, . . . , L
Z
does not depend on Y . Condi- tionally on {L = N }, Z
, L
1
, . . . , L
Z
is distributed as follows: 1. Z =
0 if N = 1; 2. Conditionally on {Z ≥
1}, L
1
is distributed as L
N
+ 1. 3. For N
′
∈ {1, . . . , N − 1}, conditionally on {Z ≥ 1, L
1
= N
′
+ 1}, Z − 1, L
2
, . . . , L
Z
is distributed as Z
, L
1
, . . . , L
Z
conditionally on {L = N
′
}. Remark
2.9. If one is interested only in the distribution of Z, L , . . . , L
Z
, L, one gets that {L
Z
, . . . , L } is distributed as {k; B
k
= 1} where B
n
, n ≥ 2 are independent Bernoulli r.v. such that PB
k
= 1 = 1
k 2
. In particular we have Z
d
= X
k≥ 2
B
k
− 1. 17
Indeed, set B
k
= 1 if the individual k, d
G
∗
at level k belongs to a fixation curve and B
k
= 0 otherwise. Notice that B
k
= 1 if none of the k − 2 look-down events which pushed the line of k, d
G
∗
between its birth time and d
G
∗
involved the line of k, d
G
∗
. This happens with probability P
B
k
= 1 =
k− 1
2 k
2
· · ·
2 2
3 2
= 1
k 2
. Moreover B
k
is independent of B
2
, . . . , B
k− 1
which depends on the lines below the line of k, d
G
∗
from the look-down construction. This gives the announced result. Notice that L = L − 1 =
787
sup{k; B
k
= 1} − 1. We deduce that conditionally on L = a, Z = P
a k=
2
B
k
with the convention Z =
0 if a = 1. In particular, we get E
[1 + λ
Z
|L = a] =
a
Y
k= 2
E [1 + λ
B
k
] =
a
Y
k= 2
kk − 1 + 2λ
kk − 1
=
a− 1
Y
k= 1
kk + 1 + 2λ
kk + 1
. The result does not change if one considers a fixed time t instead of d
G
∗
. We deduce the following Corollary from the previous Remark and Lemma 2.8 and for the first two
moments 20 we use 10 and Proposition 1.3.
Corollary 2.10. Let a ≥ 1. Conditionally on Y, L, Z
d
=
L
X
k= 2
B
k
with the convention P
;
= 0, where B
k
, k ≥ 2 are independent Bernoulli random variables independent of Y, L and such that P
B
k
= 1 = 1
k 2
. We have for all λ ≥ 0, E
[1 + λ
Z
|Y, L = a] =
a− 1
Y
k= 1
kk + 1 + 2λ
kk + 1
, 18
with the convention Q
;
= 1. We have PZ = 0|Y, L = 1 = 1 and for k ≥ 1, P
Z = k|Y, L = a = 2
k− 1
3 a +
1 a −
1 X
1a
k
···a
2
a k
Y
i= 2
1 a
i
− 1a
i
+ 2 ·
19 We also have
E [Z|Y, L = a] = 2 −
2 a
and E
[Z
2
|Y, L = a] = 18 − 4π
2
3 −
18 a
+ 8 X
k≥a
1 k
2
· 20
We are now able to give the distribution of Z conditionally on Y or X . We deduce from ii of Proposition 2.4 and from Corollary 2.10 the next result.
Corollary 2.11. Let y ∈ [0, 1]. We have, for all λ ≥ 0,
E [1 + λ
Z
|Y = y] = 1 − y
+∞
X
a= 1
y
a− 1
a− 1
Y
k= 1
kk + 1 + 2λ
kk + 1
, 21
with the convention Q
;
= 1. We have PZ = 0|Y = y = 1 − y, and, for all k ∈ N
∗
, P
Z = k|Y = y = 2
k− 1
3 1 − y
X
1a
k
···a
1
∞
a
1
+ 1a
1
+ 2 y
a
1
−1 k
Y
i= 1
1 a
i
− 1a
i
+ 2 .
22 We also have
E [Z|Y = y] = 2
1 + 1 − y
y log1 − y
. 23
The next Corollary is a direct consequence of Lemma 2.7 and Corollary 2.10 . 788
Corollary 2.12. Let x ∈ [0, 1]. We have, for all λ ≥ 0,
E [1 + λ
Z
|X = x] = x1 − x
∞
X
a= 2
x
a− 1
+ 1 − x
a− 1
+∞
X
a= 1
a− 1
Y
k= 1
kk + 1 + 2λ
kk + 1
, 24
with the convention Q
;
= 1. We have PZ = 0|X = x = 2x1 − x, and, for all k ∈ N
∗
, P
Z = k|X = x =
2
k− 1
3 x
1 − x X
1a
k
···a
1
∞
a
1
+ 1a
1
+ 2
x
a
1
−2
+ 1 − x
a
1
−2
k
Y
i= 1
1 a
i
− 1a
i
+ 2 ·
25 We also have
E [Z|X = x] = 2 1 + x logx + 1 − x log1 − x
. 26
The second moment of Z conditionally on Y resp. X can be deduced from 21 resp. 24 or from 4 and 14 resp. 16.
Some elementary computations give: P
Z = 0|X = x = 2x1 − x, P
Z = 1|X = x = 1
3
x
2
+ 1 − x
2
− 2x1 − x lnx1 − x
, P
Z = 2|X = x = 2
3 11
6 x
2
+ 1 − x
2
− 1 − x ln1 − x − x lnx +
2 3
x 1 − x
2 −
π
2
3 + 2 lnx ln1 − x −
1 3
lnx1 − x
. We recover by integration of the previous equations the following results from [26]:
P Z = 0 =
1 3
, P
Z = 1 = 11
27 and
P Z = 2 =
107 243
− 2
81 π
2
.
3 Stationary distribution of the relative size for the two oldest families
3.1 Resurrected process and quasi-stationary distribution
